Finite Group Databasebeta

← #41012367

Subgroups — #41012367

Order 27008.

282 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27213284C212yesmaximal, complemented
2#27213066C211yesmaximal, characteristic, complemented
3#27212880C212yesmaximal, complemented
4#27212872C211yesmaximal, characteristic, complemented
5#537×C211C211yesmaximal, characteristic
6C22×Dic422C2211yescharacteristic
7C2×#16809950—22no—
8C2×Dic211⋊C4—21no—
9#20269900—22no—
10C22⋊(C211⋊C8)C2212yes—
11C211⋊#95—21no—
12C211⋊#95C2211yescharacteristic
13C2×#41×C211—21no—
14C22⋊Q8×C211—22no—
15#121×C211—22no—
16#121×C211—21no—
17#121×C211C2212yes—
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×Dic422—22no—
20C2×Dic422—42no—
21C22×Dic211—21no—
22C22×Dic211—21no—
23#16809950—44no—
24C2×C211⋊C8—22no—
25Dic211⋊C4—42no—
26#41×C211—42no—
27#41×C211—42no—
28C2×Q8×C211—21no—
29C2×D4×C211—21no—
30C2×D4×C211—22no—
31C2×D4×C211—21no—
32C2×D4×C211D412yes—
33C22×C4×C211—22no—
34C22×C4×C211D411yescharacteristic
35C22×C4×C211D411yescharacteristic
36C22×C4×C211C2311yescharacteristic
37C4⋊C4×C211—21no—
38C4⋊C4×C211—22no—
39C4⋊C4×C211D412yes—
40C22⋊C4×C211—22no—
41C22⋊C4×C211—22no—
42Dic422—24no—
43Dic422—42no—
44C2×Dic211—22no—
45C2×Dic211—22no—
46C2×Dic211—41no—
47C2×Dic211—41no—
48C2×Dic211—41no—
49C2×Dic211—41no—
50C211⋊C8—42no—
51C23×C211—22no—
52C23×C211C2×D411yescharacteristic
53Q8×C211—42no—
54D4×C211—42no—
55D4×C211—42no—
56D4×C211—42no—
57D4×C211—42no—
58D4×C211—81no—
59C2×C4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211—21no—
62C2×C4×C211—22no—
63C2×C4×C211—41no—
64C2×C4×C211—44no—
65C2×C4×C211—41no—
66C2×C4×C211SD1612yescomplemented
67C2×C4×C211C2×D412yes—
68C22×C211—21no—
69C22×C211—21no—
70C22×C211—42no—
71C22×C211—42no—
72C22×C211—42no—
73C22×C211#12011yescharacteristic
74C22×C211C2×SD1611yescharacteristic
75C22×C211AGL(1,ℤ/8)11yescharacteristic
76C4×C211—22no—
77C4×C211—22no—
78C4×C211—42no—
79C4×C211—41no—
80Dic211—42no—
81Dic211—42no—
82C2×C211—21no—
83C2×C211—21no—
84C2×C211—82no—
85C2×C211#45011yescharacteristic
86C2×C211#45611yescharacteristic
87C2×C211B(4,𝔽2)11yescharacteristic
88C211#160411yescharacteristic, complemented, Sylow
89#1604—2111nomaximal, Sylow
90#537D21111yescharacteristic
91#448—2112no—
92#395—2111no—
93#327—2112no—
94#324—2111no—
95C2×#41—21no—
96C22×Q8—2111no—
97C2×SD16—4222no—
98C22⋊Q8—22no—
99#121—22no—
100#121—21no—
101#121C2×D21112yes—
102C2×C4⋊C4—4221no—
103C2×C4⋊C4C2×D21111yescharacteristic
104#103—4222no—
105C22⋊C8—2112no—
106#95—4221no—
107#95—2111no—
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×Q8—8442no—
112C2×Q8—4222no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4C22⋊D21112yes—
117C22×C4—22no—
118C22×C4—4221no—
119C22×C4—4221no—
120C22×C4C22⋊D21111yescharacteristic
121C22×C4C22⋊D21111yescharacteristic
122C22×C4C22×D21111yescharacteristic
123SD16—8444no—
124C2×C8—4222no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—8442no—
128C4⋊C4C22⋊D21112yes—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C23—22no—
132C23C2×C22⋊D21111yescharacteristic
133Q8—42no—
134Q8—8442no—
135Q8—4224no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—81no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—44no—
147C2×C4—41no—
148C2×C4—8441no—
149C2×C4—8441no—
150C2×C4—8441no—
151C2×C4—8441no—
152C2×C4—4222no—
153C2×C4—4222no—
154C2×C4#1680995012yescomplemented
155C2×C4C2×C22⋊D21112yes—
156C8—8442no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22#2026994611yescharacteristic
163C22C2×#1680995011yescharacteristic
164C22#2027000311yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—42no—
169C4—8442no—
170C4—8442no—
171C2—21no—
172C2—21no—
173C2—82no—
174C2#2721329711yescharacteristic
175C2#2721332911yescharacteristic
176C2#2721333311yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#537×C211C2
socleC22×C211#120
hypercenter#537D211
hypocenterC211#1604
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #27213284C2semidirect
2. #27213066C2semidirect
3. #27212880C2semidirect
4. #27212872C2semidirect
66. C2×C4×C211SD16semidirect
88. C211#1604semidirect
154. C2×C4#16809950semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#272132842
#272130662
#272128802
#272128722
#537×C2112
#1604211